Methods and apparatus for splitting, imaging, and measuring wavefronts in interferometry

ABSTRACT

Apparatus for splitting, imaging, and measuring wavefronts with a reference wavefront and an object wavefront. A wavefront-combining element receives and combines into a combined wavefront an object wavefront from an object and a reference wavefront. A wavefront-splitting element splits the combined wavefront into a plurality of sub-wavefronts in such a way that each of the sub-wavefronts is substantially contiguous with at least one other sub-wavefront. The wavefront-splitting element may shift the relative phase between the reference wavefront and the object wavefront of the sub-wavefronts to yield a respective plurality of phase-shifted sub-wavefronts. The wavefront-splitting element may then interfering the reference and object wavefronts of the phase-shifted sub-wavefronts to yield a respective plurality of phase-shifted interferograms. An imaging element receives and images the phase-shifted interferograms. A computer connected to the imaging element measures various parameters of the objects based on the phase-shifted interferograms. Examples of measurements include flow parameters such as the concentrations of selected gaseous species, temperature distributions, particle and droplet distributions, density, and so on. In addition to flow parameters, the displacement (e.g., the vibration) and the profile of an object may be measured.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a divisional application of U.S. patentapplication Ser. No. 10/251,729 filed Sep. 21, 2002, which applicationis a continuation of U.S. patent application Ser. No. 09/906,542 filedJul. 16, 2001, and issued as U.S. Pat. No. 6,552,808 on Apr. 22, 2003,which application is a continuation of U.S. patent application Ser. No.09/413,829 filed Oct. 6, 1999, and issued as U.S. Pat. No. 6,304,330 onOct. 16, 2001.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with Government support under Contract No.DMI-9531391 awarded by the National Science Foundation. The Governmenthas certain rights in this invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to interferometry. More particularly, thepresent invention relates to methods and apparatus for imagingwavefronts. The methods and apparatus of the present invention may beimplemented in measuring systems that measure various parameters of testobjects by simultaneously generating a plurality of phase-shiftedinterferograms.

2. Description of the Related Art

Phase-shift interferometry is an established method for measuring avariety of physical parameters ranging from the density of gasses to thedisplacement of solid objects. Interferometric wavefront sensors canemploy phase-shift interferometers to measure the spatial distributionof relative phase across an area and, thus, to measure a physicalparameter across a two-dimensional region. An interferometric wavefrontsensor employing phase-shift interferometry typically consists of aspatially coherent light source that is split into two wavefronts, areference wavefront and an object wavefront, which are later recombinedafter traveling different optical paths of different lengths. Therelative phase difference between the two wavefronts is manifested as atwo-dimensional intensity pattern known as an interferogram. Phase-shiftinterferometers typically have an element in the path of the referencewavefront which introduces three or more known phase steps or shifts. Bydetecting the intensity pattern with a detector at each of the phaseshifts, the phase distribution of the object wavefront can bequantitatively calculated independent of any attenuation in either ofthe reference or object wavefronts. Both continuous phase gradients anddiscontinuous phase gradients (speckle waves) can be measured using thistechnique.

Temporal phase shifting using methods such as piezo-electric drivenmirrors have been widely used to obtain high-quality measurements underotherwise static conditions. The measurement of transient or high-speedevents requires either ultra high-speed temporal phase shifting (i.e.,much faster than the event timescales), which is limited due to detectorspeed, or spatial phase shifting that can acquire essentiallyinstantaneous measurements.

Several methods of spatial phase shifting have been disclosed in theprior art. In 1983 Smythe and Moore described a spatial phase-shiftingmethod in which a series of conventional beam splitters and polarizationoptics are used to produce three or four phase-shifted images onto asmany cameras for simultaneous detection. A number of United Statespatents, such as U.S. Pat. Nos. 4,575,248; 5,589,938; 5,663,793;5,777,741; and 5,883,717, disclose variations of the Smythe and Mooremethod where multiple cameras are used to detect multipleinterferograms. One of the disadvantages of these methods is thatmultiple cameras are required and complicated optical arrangements areneed to produce the phase-shifted images, resulting in expensive complexsystems.

Other methods of spatial phase shifting include the use of gratings tointroduce a relative phase step between the incident and diffractedbeams, an example of which is disclosed in U.S. Pat. No. 4,624,569.However, one of the disadvantages of these grating methods is thatcareful adjustment of the position of the grating is required to controlthe phase step between the beams.

Spatial phase shifting has also been accomplished by using a tiltedreference wave to induce a spatial carrier frequency to the pattern, anexample of which is disclosed in U.S. Pat. No. 5,155,363. This methodrequires the phase of the object field to vary slowly with respect tothe detector pixels; therefore, using this method with speckle fieldsrequires high magnification.

Yet another method for measuring the relative phase between two beams isdisclosed in U.S. Pat. No. 5,392,116, in which a linear grating and fourdetector elements are used. This method has a number of drawbacks,including the inability to measure of wavefronts (i.e., the spatialphase distribution across the profile of a beam) and to form contiguousimages on a single pixilated detector such as a standard charge coupleddevice (CCD).

Finally, it is noted that wavefront sensing can be accomplished bynon-interferometric means, such as with Shack-Hartmann sensors whichmeasure the spatially dependent angle of propagation across a wavefront.These types of sensors are disadvantageous in that they typically havemuch less sensitivity and spatial resolution than interferometricwavefront sensors and are not capable of performing relative phasemeasurements such as two-wavelength interferometry.

BRIEF SUMMARY OF THE INVENTION

It is one object of the present invention to provide an interferometricwavefront sensor that incorporates spatial phase shifting but avoids thecomplexity of multi-camera systems by using a single two-dimensionalpixilated detector, such as a standard charge coupled device (CCD)camera.

It is another object of the present invention to provide methods andapparatus for performing two-wavelength interferometry that utilize acompact spatial phase-shifting device to acquire data at high speeds andprovide improved tolerance to vibration.

It is yet another object of the invention to provide methods andapparatus for dividing an incoming wavefront into four sub-wavefrontsthat are imaged substantially contiguous to maximize the coverage of apixilated area detector, while minimizing the number of necessaryoptical components to provide a compact system.

It is still another object of the invention to provide methods andapparatus for introducing a phase shift between orthogonally polarizedreference and object wavefronts that is uniform across eachsub-wavefront and not sensitive to the positioning of a diffractiveoptical element.

According to one aspect of the invention, apparatus for splitting awavefront and producing four substantially contiguous images of thewavefront consists of an input plane, a first lens element, adiffractive optical element, a second lens element, and an output plane.The lens elements are placed in a telescopic arrangement (separated bythe sum of their focal lengths) and the diffractive optical element isplaced at or near the mutual focal points. The diffractive opticalelement produces four output wavefronts (or beams) from a single inputwavefront. In a preferred embodiment the diffractive element producesfour diffracted orders of equal intensity and symmetric to the incidentaxis so that it can be characterized by a single divergence angle α anda radial angular spacing of β. The diffractive optic is constructed tosuppress the zero order component to the greatest extent possible.Alternatively, the diffractive optical element may produce threediffracted orders each of equal intensity with the transmitted zeroorder beam. The diffractive optic may include a wedged substrate toprovide a uniform angular tilt to all four beams so they propagatesymmetrically to the axis of the incident beam. Again, the compounddiffractive optical element is characterized by a single divergenceangle α and a radial angular spacing β. Any higher-order diffractedcomponents from the diffractive optic should be at least twice theangular divergence. The focal length of the second lens may be selectedto be equal to the detector size divided by two times the tangent of thediffractive optic's divergence angle. The front lens may be chosen toproduce an overall system magnification equivalent to the originalwavefront dimension divided by half the detector size.

According to another aspect of the invention, apparatus for introducinga uniform phase-shift between orthogonally polarized reference andobject wavefronts includes a polarization mask element made of discretesections. Each section includes a phase retardation plate or a blank anda linear polarizer. The relative angular orientation of the phaseretardation plate and linear polarizer is selected to be different foreach discrete section. In one exemplary embodiment, the mask elementincludes four quadrants each providing a phase shift of π/2 relative tothe clockwise adjacent quadrant.

According to still another aspect of the present invention, a system forproviding an improved wavefront sensor includes a wavefront splittingelement, a polarization mask element, a pixilated detector element, apolarization interferometer, and a computer. The phase of an object beamcan be measured with a single frame of data acquired from the pixilateddetector.

Yet another aspect of the invention provides a two-wavelengthinterferometer including a wavefront sensor with a tunable laser ormultiple laser sources. Multiple wavefronts are measured at each ofseveral wavelengths with the relative phase values subtracted todetermine the contour of an object.

Other objects, features, and advantages of the present invention willbecome apparent to those skilled in the art from a consideration of thefollowing detailed description taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is a schematic view of measurement apparatus configured inaccordance with the present invention, particularly illustrating themeasurement apparatus with the use of functional blocks;

FIG. 2 is a schematic perspective view of an exemplary embodiment ofapparatus for generating multiple phase-shifted images in accordancewith the present invention;

FIG. 3 is a schematic perspective view of an exemplary phase-retardantplate according to the invention, particularly illustrating aphase-retardant plate for shifting the phase of four wavefronts;

FIG. 4 is a plan view of the phase-retardant plate shown in FIG. 3;

FIG. 5 is a schematic view of an exemplary embodiment of measurementapparatus of the invention, particularly illustrating transmit and imageportions thereof;

FIG. 6 is a schematic view of an exemplary embodiment of an imageportion of the measurement apparatus of the invention;

FIG. 7 is a schematic view of an active surface of a detector array ofan image portion of the present invention, particularly illustrating anexemplary plurality of sub-wavefronts coaxial along an optical axis ofthe image portion;

FIG. 8 is a schematic view of another exemplary embodiment of an imagingportion of the present invention, particularly illustrating theinclusion of a polarizer and a mask;

FIG. 9 is a schematic view illustrating an exemplary imaging portion ofthe invention;

FIG. 10 is a schematic view of another exemplary embodiment ofmeasurement apparatus of the invention, particularly illustratingapparatus for performing profilometry;

FIG. 11 is a schematic view of the measurement apparatus of FIG. 6,particularly illustrating an exemplary commercial embodiment of theprofilometer of the invention;

FIG. 12 is a schematic view of a yet another exemplary embodiment ofmeasure apparatus of the invention, particularly illustrating apparatusfor measuring displacement;

FIG. 13 is a schematic view of still another exemplary embodiment of themeasurement apparatus of the invention, particularly illustratingapparatus for performing wavefront sensing; and

FIG. 14 is a schematic view of a graphical user interface illustratinginterferometric data according to the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides apparatus and methodology for measuringvarious parameters of test objects by simultaneously generating multiplephase-shifted images. More particularly, the apparatus and methodologyof the present invention enable multiple phase-shifted images (orinterferograms) to be obtained with a single imaging device and by asingle pulse of a laser and at very high rates. In doing so, the presentinvention splits, images, and measures a wavefront made up of areference and an object wavefront from an object under test.

The apparatus of the present invention may be configured to measure—insitu and in real time—flow parameters in a multiphase environment.Examples of such flow parameters include the concentrations of selectedgaseous species, temperature distributions, particle and dropletdistributions, density, and so on. In addition to flow parameters, theapparatus of the present invention may be configured to measure thedisplacement (e.g., the vibration) of an object. Moreover, the apparatusof the invention may be configured to perform profilometry of an object,that is, to measure the absolute three-dimensional profiles of solidobjects. These and other utilizations and embodiments of the technologyof the present invention are discussed in detail herein.

Turning to the drawings, a measurement system 50 exemplifying theprinciples of the present invention is illustrated in FIG. 1. Exemplarymeasurement system 50 generally includes a transmit portion 52 and animage portion 54. The transmit portion 52 transmits a referencewavefront 56 to the image portion 54 and an object wavefront 58 to anobject 60 under measurement. The reference and object wavefronts 56 and58 are preferably generated by a spatially coherent light source such asa laser. The object wavefront 58 is received by the image portion 54after acting upon the object 60, for example, by reflection or bytransmission. Data obtained by the image portion 54 from the object 60may be provided to a computer 62 for processing. The transmit portion 52and the image portion 54 may be oriented with respect to the object 60according to a plurality of measurement configurations, which arediscussed in detail below.

With continued reference to FIG. 1, exemplary image portion generallyincludes a wavefront-combining element 64 for receiving the referencewavefront 56 and the object wavefront 58 and for combining thewavefronts into a combined wavefront 66. The reference and objectwavefronts 56 and 58 are combined to be superimposed and orthogonallypolarized, which is discussed below. A wavefront-splitting element 68receives the combined wavefront 66 and splits the wavefront into aplurality of sub-wavefronts 70. A phase-shifting interference element 72receives the sub-wavefronts 70 and is configured to shift the relativephase between the reference and object wavefronts 56 and 58 and tointerfere the reference and object wavefronts 56 and 58 by polarization,for each of the sub-wavefronts 70, to yield a plurality of phase-shiftedinterferograms 74. A sensing element 76 receives the phase-shiftedinterferograms 74 from the phase-shifting interference element 72substantially simultaneously. The sensing element 76 provides data 78indicative of the interferograms 74 to the computer 62 for processing.

According to the present invention, the phase-shifting interferenceelement 72 shifts the relative phase between the reference and objectwavefronts 56 and 58 for each of the sub-wavefronts 70 discretely by afactor of a predetermined amount p. The predetermined amount p may bedetermined by a number N of sub-wavefronts 70 in the plurality ofsub-wavefronts generated by the wavefront-splitting element 68 from thecombined wavefront 66. For example, the predetermined amount p may bedetermined as the quotient of 360 degrees and the number N ofsub-wavefronts 70, or:p=360°÷N.  (1)Accordingly, the discrete phase shift Δφ of each of the plurality ofsub-wavefronts 70 may be determined as:ΔΦ_(i)=(i−1)×p,  (2)where i=1 to N. For example, if the wavefront-splitting element 68provides four sub-wavefronts 70, then the discrete phase shifts ΔΦ ofthe four wavefronts are 0°, 90°, 180°, and 270°. According to thisembodiment, there is a 90° phase shift between each of theinterferograms 74.

An exemplary embodiment of the combination of the wavefront-splittingelement 68, the phase-shifting interference element 72, and the sensingelement 76 is illustrated in FIG. 2. As shown, the combined wavefront 66includes the reference wavefront 56 from the transmit portion 52 and theobject wavefront 58 from the object 60. The wavefront-combining element64 is configured so that the reference wavefront 56 and the objectwavefront 58 are orthogonally polarized, which is indicated in FIG. 2 bythe scientific convention of an arrow and a dot. Exemplarywavefront-splitting element 68 is preferably a two-dimensionaldiffractive optical element (DOE) such as a holographic optical element(HOE) 80. According to a preferred embodiment of the invention,exemplary DOE 80 splits the combined wavefront 66 into foursub-wavefronts 70 a, 70 b, 70 c, 70 d. Each of the sub-wavefronts 70a-70 d follows a spatially discrete path.

With continued reference to FIG. 2, exemplary phase-shiftinginterference element 72 includes a plurality of sections 82, the numberof which preferably equals the number N of sub-wavefronts 70 provided bythe wavefront-splitting element 68. According to the preferredembodiment shown, exemplary phase-shifting interference element 72includes four sections 82 a, 82 b, 82 c, 82 d. The phase-shiftinginterference element 72 is disposed with respect to thewavefront-splitting element 68 so that the plurality of sub-wavefronts70 are respectively incident on the plurality of sections 82; that is,each section 82 receives one of the sub-wavefronts 70. As discussedabove, each of the sections 82 shifts the relative phase between thereference and object wavefronts 56 and 58 and interferes the twowavefronts 56 and 58 for each of the sub-wavefronts 70 incident thereonby a discrete phase shift Δφ_(i). Each of the sections 82 a, 82 b, 82 c,. . . 82N of the phase-shifting interference element 72 accordinglyprovides a respective phase-shifted interferograms 74 a, 74 b, 74 c, . .. , 74N. The phase of each phase-shifted interferogram 74 is out ofphase with the phase of the other phase-shifted interferograms 74 by afactor of the predetermined amount p of phase shift, which is discussedfurther below.

Continuing to reference FIG. 2, exemplary sensing element 76 ispreferably an imaging sensor or a detector array 84. The detector array84 may be a video-imaging sensor such as a charged coupled device (CCD)camera. According to the present invention, the detector array 84preferably has an active surface 86. The active surface 86 may bedefined by a pixel array. The detector array 84 may be made from aplurality of individual detector arrays configured to function as asingle active sensing element. For example, the active surface 86 may bedefined by more than one CCDs collectively functioning as a singlearray. For the purposes of this description, the active surface 86 has asurface area S.

The detector array 84 is disposed with respect to the phase-shiftinginterference element 72 so that the plurality of phase-shiftedinterferograms 74 are substantially simultaneously incident on theactive surface 86, thereby imaging on the active surface 86 a respectiveplurality of phase-shifted interferograms. Based on the imagedinterferograms, the spatially resolved phase of each of thephase-shifted interferograms 74 can be measured instantaneously. Inaddition, the detector array 84 is disposed with respect to thephase-shifting interference element 72 so as to maximize the area of theactive surface 86, which is discussed in more detail below.

With additional reference to FIG. 3, an exemplary embodiment of thephase-shifting interference element 72 includes a plurality of plates88. For the preferred four-component embodiment described above,exemplary phase-shifting interference element 72 includes a first plate88 a and a second plate 88 b. For purposes of clarity and illustration,the plates 88 are shown in a spaced relationship; however, according toexemplary embodiments of the invention, the plates 88 are substantiallyplanar, disposed in a parallel relationship, and abut each other. Thefirst plate 88 a includes a quarter-wave plate 90 and a blank plate 92.As known in the art, a quarter waveplate shifts the relative phase oftwo orthogonally polarized incident wavefronts by 90°, and a blank plateshifts the relative phase of two orthogonally polarized incidentwavefronts by 0° (i.e., there is no relative phase shift). The plates 90and 92 are preferably coplanar and divide the first plate 88 a intorespective halves.

The second plate 88 b of exemplary phase-shifting interference element72 includes a pair of polarizing plates 94 a and 94 b that areconfigured to polarize an incident wavefront linearly so that electricfield vectors of the transmitted wavefront are perpendicular with eachother. Specific to the illustrated embodiment, one of the polarizingplates, e.g., plate 94 a, is configured to polarize light at +45° withrespect to the vertical axis (as shown by arrow A in FIG. 3), therebyinterfering the in-phase components of the reference and objectwavefronts 56 and 58. The other polarizing plate, e.g., plate 94 b, isconfigured to polarize light at −45° with respect to the vertical axis(as shown by arrow B in FIG. 3), thereby interfering the out-of-phasecomponents of the reference and object wavefronts 56 and 58. Thepolarizing plates 94 a and 94 b are preferably coplanar and divide thesecond plate 88 b into respective halves.

With continued reference to FIG. 3 and additional reference to FIG. 4,the first and second plates 88 a and 88 b are configured so that therespective halves thereof are perpendicular with each other, thusforming a phase-retardation mask or plate 96. In the four-componentembodiment shown, the phase-retardation plate 96 includes four sections82, each of which defines a quadrant. Section 82 a, or quadrant Q₀, isdefined by the blank plate 92 and polarizing plate 94 a, thusinterfering the in-phase (i.e., 0°) component between the incidentreference and object wavefronts 56 and 58. Section 82 b, or quadrant Q₁,is defined by the quarter-wave plate 90 and polarizing plate 94 a, thusinterfering the in-phase quadrature (i.e., 90°) component between theincident reference and object wavefronts 56 and 58. Section 82 c, orquadrant Q₂, is defined by the blank plate 92 and polarizing plate 94 b,thus interfering the out-of-phase (i.e., 180°) component between theincident reference and object wavefronts 56 and 58. And section 82 d, orquadrant Q₃, is defined by the quarter-wave plate 90 and polarizingplate 94 b, thus interfering the out-of-phase quadrature (i.e., 270°)component between the incident reference and object wavefronts 56 and58.

The operation of the phase-shifting interference element 72 may bedescribed with respect to the reference and object wavefronts 56 and 58which, as mentioned above, are orthogonally polarized. The electricfield vectors for each of the wavefronts 56 and 58 may be written as:$\begin{matrix}{{\overset{\_}{E}}_{r} = {R\quad{\mathbb{e}}^{i{({{kz} - {\overset{\_}{\omega}\quad t}})}}\overset{\Cap}{s}}} & \left( {3a} \right) \\{{\overset{\_}{E}}_{s} = {S\quad{\mathbb{e}}^{i{({{kz} - {wt} + {\Delta\phi}})}}\overset{\Cap}{p}}} & \left( {3b} \right)\end{matrix}$where:

R and S are the amplitudes of each wavefront 56 and 58, respectively;

is the optical frequency;

t is time;

k is the wavevector=2π/λ;

p and s are orthogonal unit polarization vectors; and

ΔΦ is the phase difference between the wavefronts 56 and 58.The intensity (I) of each of the phase-shifted interferograms 74incident on the active surface 86 of the detector array 84 is given by:$\begin{matrix}{I_{0} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos({\Delta\phi})}}} \right)}} & \left( {4a} \right) \\{I_{1} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos\left( {{\Delta\phi} + \frac{\pi}{2}} \right)}}} \right)}} & \left( {4b} \right) \\{I_{2} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos\left( {{\Delta\phi} + \pi} \right)}}} \right)}} & \left( {4c} \right) \\{I_{3} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos\left( {{\Delta\phi} + \frac{3\pi}{2}} \right)}}} \right)}} & \left( {4d} \right)\end{matrix}$where I_(r) and I_(s) are the intensities of the reference and objectwavefronts 56 and 58, respectively (which intensities are proportionalto R² and S²). This set of phase-shifted intensities I₀, I₁, I₂, and I₃may be analyzed numerically using a number of algorithms to solveexplicitly for the phase difference between the reference and objectwavefronts 56 and 58, which is discussed in detail below.

As it is preferable to maximize the imaging area of the detector array84 (i.e., to maximize the portion of the surface area S of the activesurface 86 that is illuminated by the interferograms 74), thephase-retardation plate 96 is preferably disposed adjacent to orsubstantially at the active surface 86 of the detector array 84, whichis discussed in more detail below. By detecting the plurality ofphase-shifted interferograms 74 instantaneously with an imaging sensorexemplified by the detector array 84, the image portion 54 of theinvention enables the measuring system 50 to instantaneously measure theentire test object 60. In addition, the instantaneous detection of thephase-shifted interferograms 74 eliminates the need to scan individualbeams spatially through or across the surface of the object 60.

As mentioned above, exemplary measurement system 50 of the presentinvention may be configured in a plurality of preferred embodiments eachdesigned to carry out a particular type of real-time measurement,including a profilometer, a displacement sensor, and a wavefront sensor.In other words, exemplary embodiments of the measuring system 50 includea common transmit portion 52 and a common image portion 54 that can bephysically oriented in a plurality of configurations with a plurality ofoptical and imaging components to undertake a plurality of measurements,which is discussed in detail below.

FIG. 5 illustrates one such exemplary configuration of the measurementsystem 50 of the invention which may be used to perform real-timeinterferometry for measuring transient events. The transmit portion 52according to this embodiment includes a coherent light source such as alaser or laser diode 98. The laser 98 may include a half-wave plate 100to provide a coherent light wavefront 102 which is split by a polarizingbeam splitter (PBS) 104 into the reference wavefront 56 and the objectwavefront 58. The PBS 104 is configured to provide orthogonallypolarized wavefronts as shown. The object wavefront 58 is expanded by,for example, a combination of an expanding lens 106 and a collimatinglens 108. Upon expansion, the object wavefront 58 is transmitted to thetest object 60 where the object wavefront 58 is incident upon thesurface or boundary thereof and either reflected from or transmittedthrough the object 60.

Exemplary image portion 54 receives the object wavefront 58 from theobject 60 and may include optics for collimating the received objectwavefront 58, such as a combination of a collecting lens 110 and acollimating lens 112. Collimating lens 112 is preferably spaced from thecollecting lens 100 by a distance equal to the sum of their respectivefocal lengths f₁ and f₂. The object wavefront 58 is then superimposedwith the reference wavefront 56 at the wavefront-combining element 64which may be a polarizing beam splitter (PBS) 114 to yield the combinedwavefront 66. PBS 114 is preferably spaced from collimating lens 112 bya focal length f₂ of the collimating lens. The combined wavefront 66 maybe focused on the diffractive optical element 80 by means of a convexlens 116. In turn, the plurality of sub-wavefronts 70 may be focused onthe phase-retardation/interference plate 96 either directly or by meansof a collimating lens 118 as shown.

The placement of the various elements with respect to each other ischosen to maximize the operability of the image portion 54. For example,PBS 114, the convex lens 116, and the diffractive optical element 80 arepreferably respectively spaced apart by focal length f₃, which is thefocal length of the convex lens 116. In addition, the diffractiveoptical element 80, the collimating lens 118, and thephase-retardation/interference plate 96 are preferably respectivelyspaced apart by a focal length f₄, which is the focal length of thecollimating lens 118. The placement of the diffractive optical element80 at the focus of collimating lens 118, which is defined as the inputfocal plane or the Fourier transform plane, optimizes the area of theactive surface 86 of the detector array 84 illuminated by the pluralityof phase-shifted interferograms 74.

Referencing FIG. 6, the optics of exemplary imaging portion 54 are shownin more detail. The optical elements of the imaging portion 54 arealigned along an optical axis O. As mentioned above, the diffractiveoptical element 80 splits the combined wavefront 66 into a plurality of(e.g., four) sub-wavefronts 70. Each of the sub-wavefronts 70 follows anoptical path defined by the distance each of the sub-wavefronts 70follows from the diffractive optical element 80 to the active surface 86of the detector array 84.

The diffractive optical element 80 and lenses 116 and 118 are configuredso that each of the imaged sub-wavefronts 70 incident at detectorsurface 86 are adjacent to or substantially contiguous with at least oneother sub-wavefront, which is shown in FIG. 7. For example, in theexemplary embodiment shown, sub-wavefront 70 a is substantiallycontiguous with sub-wavefronts 70 b and 70 c, which is respectivelyindicated by reference alphas AB and AC; sub-wavefront 70 b issubstantially contiguous with sub-wavefronts 70 a and 70 d, which isrespectively indicated by reference alphas AB and BD; sub-wavefront 70 cis substantially contiguous with sub-wavefronts 70 a and 70 d, which isrespectively indicated by reference alphas AC and CD; and sub-wavefront70 d is substantially contiguous with sub-wavefronts 70 b and 70 c,which is respectively indicated by reference alphas BD and CD. Thissubstantially contiguous nature of the sub-wavefronts 70 is furtherenhanced in an embodiment in which the diffractive optical element 80splits the combined wavefront 66 into a plurality of sub-wavefrontshaving a substantially rectangular cross section as shown in FIG. 8.

The exemplary diffractive optical element 80 preferably splits thecombined wavefront 66 in such a manner that the sub-wavefronts 70diverge from the optical axis O at substantially equal angles. In apreferred embodiment, the diffractive optical element 80 may producefour diffracted orders that have equal intensity and are symmetric tothe incident axis so that the diffracted orders may be characterized bya single divergence angle α and a radial angular displacement β. Thediffractive optical element 80 may be constructed to suppress the zeroorder component to the greatest extent possible.

In another exemplary embodiment, the diffractive optical element 80 mayproduce three diffracted orders each of equal intensity with thetransmitted zero order beam. The diffractive optical element 80 mayinclude a wedged substrate to provide a uniform angular tilt to all fourbeams so that the beams propagate symmetrically to the axis of theincident beam. As mentioned above, the diffractive optical element 80 ispreferably characterized by a single divergence angle α and a radialangular displacement β.

Referring to FIG. 7, the radial angular displacement β produced byexemplary diffractive optical element 80 is determined by the aspectratio of the height h and the width w of the active surface 86 of thedetector array 84. The desired radial angular displacement β is givenby: $\begin{matrix}{\beta = {2\quad{\tan^{- 1}\left( \frac{h}{w} \right)}}} & (5)\end{matrix}$where w and h are the width and the height of the active surface 86 ofdetector array 84. For a detector with a unity aspect ratio (i.e.,square), the radial angular displacement β becomes 90 degrees and allfour images are radially symmetric.

Accordingly, each of the sub-wavefronts 70 follows an independentoptical path from the diffractive optical element 80 to the activesurface 86 that has a length substantially equal to each of the otheroptical paths. As such, the plurality of sub-wavefronts 70 reach theactive surface 86 substantially simultaneously. By configuring theimaging portion 54 so that the sub-wavefronts 70 have substantiallyequal optical path lengths, the imaging portion 54 is less susceptibleto errors that may introduced by vibration to the system.

With particular reference to FIG. 7, exemplary active surface 86 of thedetector array 84 may have a plurality of sections 119 for respectivelyreceiving the plurality of sub-wavefronts 70. Each of the sections 119has a surface area on which the respective sub-wavefront 70 is incident.According to the present invention, the portion or percentage of thesurface area of each section 119 on which a sub-wavefront is incident ispreferably maximized, thereby maximizing the resolution of the detectorarray 84. For example, each of the sub-wavefronts 70 a-70 d is incidenton at least half of the surface area of a respective section 119 a-119d. More preferably, the percentage is at least 75%. In the embodimentshown in FIG. 7 by the circular cross-hatched regions, the incidentpercentage of each sub-wavefront 70 may be determined by πr² divided by(h/2+w/2)². In the embodiment shown in FIG. 7 by the rectangular crosshatched region, the incident percentage of each sub-wavefront issubstantially 100%.

Further referencing FIG. 6 and with addition reference to FIG. 8, anaperture 121 may be provided at an input focal plane of the convex lens116 (i.e., at a focal length f₃), with the diffractive optical element80 positioned at the output focal plane of the convex lens 116.Alternatively, as shown in FIG. 9, a pair of apertures 121 a and 121 bmay be positioned upstream of PBS 114 through which the reference andobject wavefronts 56 and 58 respectively travel. According to apreferred embodiment of the invention, the aperture(s) 112 may berectangular with an aspect ratio substantially the same as the activesurface 86 of the detector array 84. The presence of the aperture(s) 121reduces the amount of ambient noise received in the image portion 54 andreduces crosstalk between the imaged sub-wavefronts.

An example of a design method that maximizes the surface area coveragefollows. With reference to FIGS. 6 and 7, the focal length of lens 118is selected to be equal to one fourth of the diagonal length D of theactive area of detector 84 divided by the tangent of the divergenceangle α of the diffractive optical element 80. For illustrative clarity,the diagonal length D is shown as segment AB in FIG. 7. Thus:$\begin{matrix}{f_{4} = \frac{D}{4\quad\tan\quad\alpha}} & (6)\end{matrix}$The front lens 116 is chosen to produce an overall system magnificationequivalent to the diagonal length d_(i) of the input aperture 112 (shownin FIG. 6) divided by the diagonal length D of the detector array 84.Thus: $\begin{matrix}{f_{3} = {\frac{d_{i}}{D}f_{4}}} & (7)\end{matrix}$The overall length L of the imaging portion 54 is given by:$\begin{matrix}{L = {{2\left( {f_{3} + f_{4}} \right)} = \frac{\left( {d_{i} + D} \right)}{2\quad\tan\quad\alpha}}} & (8)\end{matrix}$

According to an exemplary embodiment of the invention, the aperture(s)121 may be selected so that the diagonal length d_(i) is substantiallyequal to the diagonal length D of the detector array 84 (i.e., d_(i)=D).According to such an embodiment, focal length f₃ is equal to focallength f₄ and the overall system length L is given by: $\begin{matrix}{L = {{2\left( {f_{3} + f_{4}} \right)} = \frac{D}{\tan\quad\alpha}}} & (9)\end{matrix}$It can be seen from Equations 7 and 8 that in many embodiments it isdesirable to have a large diffractive optic divergence angle α to reducethe overall size of imaging portion 54. In practice, divergence angles αof 5 degrees to 10 degrees produce a relatively compact system.

In addition to the real-time interferometer embodiment illustrated inFIG. 5, exemplary measurement system 50 of the present invention may beconfigured in a plurality of additional preferred embodiments eachdesigned to carry out a particular type of real-time measurement,including a profilometer, a displacement sensor, and a wavefront sensor,each of which is described in detail below.

Referencing FIG. 10, exemplary measurement system 50 of the presentinvention is configured to perform profilometry. Exemplary profilometer50 is configured to perform on-axis illumination and viewing, which isuseful in obtaining three-dimensional (3D) information of the object 60.Many industries utilize profilometry in research and development,quality control, and manufacturing, including the semiconductor andmedical industries.

Exemplary transmit portion 52 includes the laser 98 which transmits thecoherent light wavefront 102. A single polarizing wavefront splitter(PBS) 120 is shared by both the transmit and image portions 52 and 54for splitting the light wavefront 102 into the reference wavefront 56and the object wavefront 58 and combining the reference wavefront 56 andthe object wavefront 58 into the combined wavefront 66. In addition toPBS 120, exemplary image portion 54 of the profilometer includes theconvex lens 116, the diffractive optical element 80, the collimatinglens 118 displaced from element 80 by its focal length, thephase-retardation/-interference plate 96, and the CCD camera. Thecomputer 62 may be connected to both the transmit and image portions 52and 54 to control the operation of the laser 98 and to receive imagingdata 78 from the detector array 84.

FIG. 11 illustrates an exemplary commercial embodiment of theprofilometer 50 of FIG. 10. As shown, the laser 98 provides the lightwavefront to an integrated measuring unit 122 by means of an opticalcable 124. The integrated measuring unit 122 includes a housing 126 inwhich the common PBS 120, as well as each of the elements of the imageportion 54 shown in FIG. 9, is received. The integrated measuring unit122 transmits and receives the object wavefront 58, with the detectorarray 84 providing image data to the computer 62 via a cable 128.

Referencing FIG. 12, another exemplary commercial embodiment of themeasurement system 50 of the present invention is shown and configuredto function as a displacement sensor. Displacement sensors are useful inmeasuring, for example, the vibration or the strain of an object.Exemplary transmit portion 52 of the displacement-sensor embodiment ofthe measuring system 50 includes the laser 98 which transmits thecoherent light wavefront to a fiber wavefront splitter 130 via anoptical cable 132. The fiber wavefront splitter 130 splits the lightwavefront into the reference wavefront 56, which is provided to theimage portion 52 by an optical cable 134, and the object wavefront 58,which is provided to an optics unit 136 by an optical cable 138. Theoptical unit 136 of the transmit portion 52 includes thewavefront-expanding optics of the concave lens 106 and collimating lens108 (see FIG. 5). The operation of the displacement sensor illustratedin FIG. 12 is analogous to that described above.

According to the displacement-sensor embodiment of the measurement unit50, the separate and portable optics unit 136 may be positioned relativeto the test object 60 and the image portion 54. The object wavefront 58can thus be directed to the object 60 from any angle or position.

Referencing FIG. 13, yet another exemplary commercial embodiment of themeasurement system 50 of the present invention is shown and configuredto function as a wavefront sensor. Wavefront sensors may be used tomeasure, for example, pressure, temperature, or density gradients intransparent solids, liquids, or gases. Exemplary transmit portion 52 mayinclude an integrated transmit unit 140 with a housing 142, andexemplary image portion 54 may include an integrated receive unit 144with a housing 146. Similar to the layout of the measurement system 50shown in FIG. 5, exemplary transmit unit 140 of the wavefront-sensorembodiment of the measuring system 50 includes the laser which transmitsthe reference wavefront 56 to the integrated receive unit 144 via anoptical cable 148 and the object wavefront 58 to the test object 60. Theoperation of the wavefront sensor illustrated in FIG. 13 is analogous tothat described above.

For each of the foregoing embodiments of the measuring system 50 of thepresent invention, a software application may be utilized by thecomputer 62 for data acquisition and processing. The softwareapplication causes the computer 62 to acquire, process, analyze, anddisplay data associated with the phase-shifted interferograms 74. Dataacquisition may be accomplished by recording two interferograms for eachmeasurement: a reference interferogram for the reference wavefront 56and an object interferogram for the object wavefront 58. Wrapped phasemaps are calculated for each of the interferograms and then subtractedfrom each other. The result is unwrapped to yield a map of the phasechange between the reference and object interferograms. Unwrapping isthe procedure used to remove the modulo 2π ambiguity that ischaracteristic of interferometric data.

Phase may be calculated based on a single frame of data according to:Φ(x,y)=tan⁻¹ {[I ₃(x,y)−I ₁(x,y)]÷[I ₀(x,y)−I ₂(x,y)],  (10)where I₀, I₁, I₂, and I₃ are the respective intensities of each of thephase-shifted interferograms 74 a-74 d incident on the active surface 86of the detector array 84 from the four sections 82 a-82 d (i.e.,quadrants Q₀, Q₁, Q₂, and Q₃) as calculated in Equations 4a-4d above.The variables x and y are the pixel coordinates. To reduce noise in theimage, spatial averaging may be used to smooth the phase map whileretaining a sharp transition at the 2π-0 phase step. The spatiallyaverages phase may be calculated using the following equations:Φ(x,y)=tan⁻¹{sum(x,yεδ)[I ₃(x,y)−I ₁(x,y)]÷sum(x,yεδ)[I ₀(x,y)−I₂(x,y)]},  (11)where the sums are performed over the range of δ nearest neighbors.Increasing the number of averaged pixels improves smoothness of thephase map at the expense of spatial resolution; however, the sharpnessof the phase discontinuity is retained, thereby permitting rapid phaseunwrapping. The unwrapping of phase maps removes the discontinuous stepand permits quantitative analysis of the images.

The number of pixels averaged may be selected by a user. For comparingtwo states of the system of to subtract background phase noise from thesystem, the phase difference mode can be used. Phase may be calculatedaccording to:ΔΦ(x,y)=tan⁻¹ [X(x,y)÷Y(x,y)],  (12)where:

-   -   X(x,y)=[Ib₃(x,y)−Ib₁(x,y)]*[It₀(x,y)−It₂(x,y)]−[It₃(x,y)−It₁(x,y)]*[Ib₀(x,y)−Ib₂(x,y)],    -   Y(x,y)=[Ib₀(x,y)−Ib₂(x,y)]*[It₀(x,y)−It₂(x,y)]+[Ib₃(x,y)−Ib₁(x,y)]*[It₃(x,y)−It₁(x,y)],    -   Ib is the baseline image captured, and    -   It is the image captured for comparison.        Spatial averaging can be accomplished using the formula:        ΔΦ(x,y)=tan⁻¹[sum(x,yεδ)X(x,y)÷sum(x,yεδ)Y(x,y)].  (13)

The three dimensional shape of an object can be determined by using twocolor interferometry. To do so, a first set of four phase-shiftedinterferograms is captured at a first wavelength λ₁ (i.e., Ib_(n)), anda second set of phase-shifted interferograms is captured at a secondwavelength λ₂ (i.e., It_(n)). The relative distance to the object (orrange) is calculated by: $\begin{matrix}{{{R\left( {x,y} \right)} = {\frac{\lambda^{2}}{4\quad{\pi\Delta\lambda}}{\tan^{- 1}\left( \frac{X\left( {x,y} \right)}{Y\left( {x,y} \right)} \right)}}},} & (14)\end{matrix}$where:

-   -   X(x,y)=[Ib₃(x,y)−Ib₁(x,y)]*[It₀(x,y)−It₂(x,y)]−[It₃(x,y)−It₁(x,y)]*[Ib₀(x,y)−Ib₂(x,y)]    -   Y(x,y)=[Ib₀(x,y)−Ib₂(x,y)]*[It₀(x,y)−It₂(x,y)]+[Ib₃(x,y)−Ib₁(x,y)]*[It₃(x,y)−It₁(x,y)]

Noise in the image can be significantly reduced using a weighted spatialaverage over neighboring pixels. This can be accomplished by:$\begin{matrix}{{{R\left( {x,y} \right)} = {\frac{\lambda^{2}}{4\quad{\pi\Delta\lambda}}{\tan^{- 1}\left( \frac{\sum\limits_{x,{y \in \delta}}{X\left( {x,y} \right)}}{\sum\limits_{x,{y \in \delta}}{Y\left( {x,y} \right)}} \right)}}},} & (15)\end{matrix}$where the sums are performed over the range of δ nearest neighbors.Because of the modelo 2π behavior of the arctangent function, the rangeis wrapped (ambiguous) beyond the so-called synthetic wavelength of:$\begin{matrix}{\lambda_{s} = {\frac{\lambda^{2}}{4\quad{\pi\Delta\lambda}}.}} & (16)\end{matrix}$

The well-known process of spatial phase unwrapping can be used to removethe discontinuous steps and to permit quantitative analysis of theimages. Alternatively, it is possible to use multiple syntheticwavelengths and incrementally add the range distance as known in theart. The overall range is then given by: $\begin{matrix}{{{R^{\prime}\left( {x,y} \right)} = {\sum\limits_{m}\frac{R_{\Delta\quad\lambda\quad m}\left( {x,y} \right)}{m}}},} & (17)\end{matrix}$where m is the number of wavelength steps used and R_(Δλm) is the rangemeasured with a frequency tuning of Δλ/m. Implied in this method is thatno single measurement should have a phase value greater than 2π, whichcan place a restriction on the maximum size of the object that can bemeasured.

Referencing FIG. 14, a user interface 148 provided by the software ofthe invention is shown displaying a raw interferogram 150 and wrappedphasemaps 152 from a central portion of the raw interferogram 150. Theraw interferogram 150 illustrates data 78 resulting from the measurementof a diffusion flame.

Those skilled in the art will understand that the preceding exemplaryembodiments of the present invention provide the foundation for numerousalternatives and modifications thereto. These other modifications arealso within the scope of the present invention. Accordingly, the presentinvention is not limited to that precisely as shown and described above.

1. A method for sensing a wavefront with a sensing element including anactive surface having a plurality of sections each with a surface area,the method comprising splitting the wavefront into a plurality ofsub-wavefronts wherein at least one of the sub-wavefronts illuminates atleast 80% of the surface area of a respective one of the sections of theactive surface.
 2. The method of claim 1 wherein the wave front is splitso that at least one of the sub-wavefronts illuminates substantially100% of the surface area of a respective one of the sections of theactive surface.
 3. The method of claim 2 wherein the wavefront is splitinto four sub-wavefronts and each of the sub-wavefronts illuminatessubstantially 100% of the surface area of a respective one of thesections of the active surface.
 4. The method of claim 1 furthercomprising simultaneously measuring the phase of each of thesub-wavefronts.
 5. Apparatus for sensing a wavefront including areference wavefront and an object wavefront, the apparatus comprising: awavefront-splitting element for splitting the wavefront into a pluralityof sub-wavefronts; a phase-shifting interference element including aplurality of quadrants each for shifting the phase of a respective oneof the sub-wavefronts by a multiple of 90° including 0°; and a sensingelement including an active surface having a plurality of sections;wherein each of the sub-wavefronts follows an optical path from thewavefront-splitting element to a respective one of the quadrants of thephase-shifting element to a respective one of the sections of the activesurface of the sensing element; and wherein each of the quadrantsincludes two plates such that each of the optical paths has asubstantially equal path length.
 6. The apparatus of claim 5 wherein atleast one of the sub-wavefronts is incident on at least 80% of thesurface area of a respective one of the sections of the active surface.7. The apparatus of claim 6 wherein each of the phase-shiftedinterferograms is incident on substantially 100% of the surface area ofa respective one of the sections.
 8. The apparatus of claim 5 wherein atleast one of the sub-wavefronts is adjacent to another one of thesub-wavefronts at the active surface.